The New York Giants have more or less completed the first wave of free agency, adding 10 new players while re-signing 10 from last season.

Their work, though, is far from over, as now they begin figuring out their board for the annual NFL draft on May 8-10.

The Giants, who arrange their draft board into rows, will continue to fine-tune the grades on certain players probably right up until the start of the draft.

In trying to project which players are rising versus which are falling, I’ve considered how much a need still exists at that position and have looked at any notable results from the players’ respective pro days and combine performances.

Barr’s long arms would also be beneficial in pass defense, especially if the Giants are looking for their defensive front to get their hands on more passes.

Based on the annual year-end stats provided by the team, the number of pass breakups by defensive linemen has declined, from 20 in 2011, their Super Bowl season, to 14 last season.

If drafted, he would presumably fill that "Joker" position that the Giants had envisioned for Damontre Moore last year. However, with Justin Tuck having left for Oakland, a healthy Moore will probably compete with Mathias Kiwanuka for Tuck’s old job.

Barr’s ability to not only rush the passer but also hold his own in coverage is something that the Giants haven’t had from a starting linebacker for quite some time now. That's a good reason for his stock to be rising on their draft board.

Besides reports of him not impressing in drills, his receiving ability, which is a strength, didn’t draw rave reviews, based on reports.

He is not known for his blocking, which despite the gravitation toward tight ends who can function as receivers in the offense, is still very much part of the job description, at least judging by what the Giants have done in their running game in the past.

Maybe it was just a matter of it being a bad day, but if head coach Tom Coughlin really wants to get back to Giants football, per Bart Hubbuch of the New York Post, presumably the Giants will want more out of their tight end than just a receiver, especially considering that their offensive line remains a work in progress.

Rising: Tight End Arthur Lynch, Georgia

The more I think about it, the more I believe the Giants will do two things at the tight end spot.

The first is to try to add a veteran. It’s possible that they could be waiting to see what happens with Green Bay’s Jermichael Finley, an unrestricted free agent who, per ESPN.com, still hasn’t been cleared by doctors to return to football four months removed from single fusion surgery to the C-3/C-4 vertebra in his neck.

As general manager Jerry Reese noted in a radio interview last month with WFAN, a player’s recovery from neck and/or back injuries can be unpredictable.

The second option is the draft. I had thought that perhaps Notre Dame’s Troy Niklas, rated as the fourth-best tight end prospect by CBSSports.com, might be of interest, but the Giants would probably have to spend at least a second-round pick to get him.

Instead, I think the Giants might wait until Day 3 to add a tight end. A candidate who is known for his blocking is Georgia’s Arthur Lynch, whom CBSSports.com projects can be had in the fourth round.

He doesn’t have the speed and quickness of Ebron in the receiving game, but Rob Rang notes, “Lynch shows reliable hands and has a big frame to shield defenders from the ball.”

Rang also notes that the 6’5”, 258-pound senior “is particularly effective on combination blocks in which he initially helps an offensive tackle double-team a defensive lineman before switching off to chip a linebacker at the second level.”

Lynch finished the 2013 season with 30 receptions for 459 yards and five touchdowns for the Bulldogs, all career highs, so he can contribute in the passing game.

However, it's not known just what kind of role new offensive coordinator Ben McAdoo envisions for the tight end, so it's probably a safe bet to assume that whoever the next tight end is, he'd have to be more than decent as a blocker and as a receiver.

Falling: Offensive Lineman Taylor Lewan, Michigan

Back in December, Giants CEO John Mara admitted to reporters the obvious: That the team took a few chances on some players with the hope that if they were right, “maybe we’ll knock it out of the park.”

When he was later asked if the team would consider taking fewer chances on college prospects with extensive injury histories or off-field issues, he said, “Possibly, depending. Certainly you want guys coming in there healthy and don’t have any off-field issues, but that’s not necessarily what you get coming out of the colleges and sometimes you have to take those risks.”

It will be interesting to see if the Giants are willing to make an exception and take a risk on Michigan offensive tackle Taylor Lewan (6’7”, 309 pounds).

Per a report by USA Today, he will be arraigned on April 8 on three charges related to a fight he was involved in following a game on Dec. 1.

Besides the pending legal issues that make the selection of Lewan a risk for the Giants, who desperately need to start hitting on more of their draft picks starting this year, there has been nothing to suggest that incumbent Will Beatty, who is recovering from a broken leg suffered in the regular-season finale, won’t be ready for the 2014 season.

Thus between the likely return of Beatty and the uncertainty regarding Lewan's immediate future, the Giants might decide to pass on this prospect.

Rising: Center Bryan Stork, Florida State

Despite their acquisition of J.D. Walton, the center position is far from being settled.

Forget that he has played in just 20 games for Denver since 2011 and had an injury issue that cost him most of the 2012 and 2013 seasons.

Right now the Giants have no depth to speak of at the position thanks to their decision to move on from Jim Cordle.

It would not be a stretch to assume that the Giants have their eye on a young center whom they can begin grooming this year to be ready by the time Walton's two-year contract expires.

One such prospect who could use a year or two of seasoning is Bryan Stork (6’4", 315 pounds).

Stork, whom CBSSports.com compares to Saints unrestricted free agent Brian De La Punte, is a solid technician who has played with good leverage. However, he’s in need of additional strength work in the lower half of his body.

If the Giants can get Stork on Day 3, they can bring him along, while Walton mans the position this year. In addition, Stork, a former high school tight end, could probably get on the field as the blocking tight end, assuming the Giants plan to run the jumbo package under McAdoo.

With Rodgers-Cromartie and Prince Amukamara set to be the starting duo at cornerback and Thurmond projected to be the slot corner, the Giants no longer have a pressing need at this position.

That’s why someone like Justin Gilbert (6’0”, 200 pounds) of Oklahoma State might not be as high up on the priority list.

He is still very much a first-round prospect and will be an exceptional player if he stays healthy. Because of the Giants’ approach to free agency, they probably won't select him. If faced with making a choice between Gilbert and others who might have a similar grade, they will likely go in a different direction.

Rising: Defensive End Ben Gardner, Stanford

If the Giants are looking to take a flier on a defensive end prospect late in the draft, Stanford’s Ben Gardner, who was not invited to the NFL Scouting Combine last month, might be worth some serious consideration.

He (6’4”, 262 pounds) had his 2013 season cut short thanks to a left pectoral injury. Prior to his injury, he had racked up 7.5 tackles for a loss and 4.5 sacks.

During Stanford’s pro day, Gardner, who because of his size is regarded as a ‘tweener, had an impressive showing.

CBSSports.com's Rob Rang noted that Gardner has “shown the grit to handle remaining on the defensive line as well as enough fluidity and instincts to operate as a linebacker.”

With the Giants' attempt at filling the "Joker" role with O'Brien Schofield having failed reportedly due to concerns about his knee, per NFL.com's Ian Rapoport, if New York passes on UCLA’s Anthony Barr on Day 1, Gardner might be the next best option.

However, Benjamin, whom Rang compares to former Giants receiver Plaxico Burress, is also rough around the edges in that he doesn't always run crisp routes and has had some issues with dropped passes when he takes his eyes off the ball.

The Giants are probably going to want a more sure-handed receiver, and with the class being so deep, Benjamin is not a sure thing.